The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part I: iterative and explicit analytical solution for non-linear transverse vibrations

2003 ◽  
Vol 265 (1) ◽  
pp. 123-154 ◽  
Author(s):  
M. Haterbouch ◽  
R. Benamar
Keyword(s):  
Analytical Solution ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Circular Plates ◽  
Axisymmetric Mode ◽  
Transverse Vibrations ◽  
Explicit Analytical Solution ◽  
Non Linear

Non-linear free transverse vibration of thin rotating discs

2007 ◽  
Vol 221 (3) ◽  
pp. 467-473 ◽  
Author(s):  
H R Hamidzadeh
Keyword(s):  
Steady State ◽  
Free Vibration ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Stress Distributions ◽  
Rotating Speed ◽  
Rotating Discs ◽  
Non Linear ◽  
Modes Of Vibration ◽  
Free Transverse Vibration

An analytical method is adopted to determine modal characteristics of non-linear spinning discs. The disc is assumed to be isotropic and rotating under steady-state conditions. The effects of amplitude and rotating speed on natural frequencies are determined. The developed procedure is also capable of analysing natural frequencies of linear free vibration, which is independent of amplitude. Attention is confined to determine natural frequencies, mode shapes, stress distributions, and critical speeds for different numbers of nodal diameters. The developed procedure does not consider modes of vibration corresponding to nodal circles. Validity of this procedure is verified by comparing some of the computed results with those established for certain cases.

Analytical Solution for Vibration of a Rotating Delaminated Composite Beam with End Mass

2016 ◽  
Vol 16 (06) ◽  
pp. 1550013 ◽  
Author(s):  
Ramazan-Ali Jafari-Talookolaei
Keyword(s):  
Analytical Solution ◽  
Natural Frequencies ◽  
Laminated Composite ◽  
Mode Shapes ◽  
Material Anisotropy ◽  
Rotating Speed ◽  
Multipliers Method ◽  
Free Mode ◽  
The Difference ◽  
Laminated Composite Beams

In this paper, the free vibration of rotating laminated composite beams (LCBs) with general lay-ups and single through-the-width delamination is analytically investigated. The Hamilton’s principle is used to derive the coupled governing differential equations and boundary conditions for the rotating delaminated beam, considering the effects of shear deformation, rotary inertia, material couplings (bending–tension, bending–twist and tension–twist couplings), and Poisson’s effect. Both the free mode and constrained mode assumptions are adopted. Analytical solution for the natural frequencies and mode shapes are presented by incorporating the constraint conditions using the Lagrange multipliers method. The accuracy is assured by the convergence of the natural frequencies, as well as by comparison with published results. The effects of various factors such as delamination parameter, fiber angle, hub radius, material anisotropy, end mass and rotating speed are studied in detail. The difference between the results based on the free mode and constrained mode assumptions is also investigated.

Free Vibration of In-Planar Rotating Ring: An Analytical Solution

2000 ◽  
Author(s):  
Hamid R. Hamidzadeh
Keyword(s):  
Analytical Solution ◽  
High Speed ◽  
Systematic Approach ◽  
Natural Frequencies ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Stress Theory ◽  
Wave Numbers ◽  
Annular Disks ◽  
Circumferential Wave

Abstract An analytical method is presented to consider the vibration characteristics of high speed rotating rings. More specifically, a systematic approach based on an established solution for linear in-plane vibration of spinning annular disks is used to compute natural frequencies and mode shapes of rotating rings. The medium is considered to be homogenous, and elastic isotropic. The developed analytical solution is achieved by implementing the two-dimensional plane stress theory. The modal displacements and stresses at both inner and outer boundaries are determined, and the required boundary conditions are satisfied to obtain the frequency equation. The dimensionless natural frequencies for different modes, rotating speeds, and thickness ratios are computed. In addition, variations of dimensionless critical speeds for several circumferential wave numbers versus radius ratio of the ring are presented. The provided results are for the two different cases of clamped-free and free-free rings.

VIBRATION AND STABILITY OF A HEAVY AND HEATED VERTICAL CIRCULAR PLATE

2010 ◽  
Vol 10 (05) ◽  
pp. 1111-1121 ◽  
Author(s):  
R. MARETIC ◽  
V. GLAVARDANOV ◽  
V. MILOSEVIC-MITIC
Keyword(s):  
Circular Plate ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Plate Edge ◽  
Asymmetric Mode ◽  
Critical Weight ◽  
Transverse Vibrations ◽  
Weight Parameter ◽  
Plate Stability ◽  
Parameter Values

This paper is concerned with the vibration and stability of a standing, heavy, and circular plate when heated. This investigation also deals with the plate being exposed to inertial forces due to uniform acceleration. The plate edge is clamped. Natural frequencies of transverse vibrations depending on the plate weight and temperature were determined using the Galerkin's method. Mode shapes are given for some frequencies including the influence of weight parameter on changes in mode shapes. It is shown that frequencies split for the asymmetric mode shapes, so that there are two different frequencies in those cases. Critical weight parameter values where plate stability ceases were determined. Critical values of the weight parameter depending on Poisson's ratio are also presented herein.

scholarly journals Vibration analysis of submerged rectangular microplates with distributed mass loading

2009 ◽  
Vol 465 (2104) ◽  
pp. 1323-1336 ◽  
Author(s):  
Zhangming Wu ◽  
Xianghong Ma ◽  
Peter N Brett ◽  
Jinwu Xu
Keyword(s):  
Analytical Solution ◽  
Natural Frequencies ◽  
Isotropic Plate ◽  
Numerical Results ◽  
Previous Literature ◽  
Mode Shapes ◽  
Mass Loading ◽  
Coupling System ◽  
Different Types ◽  
Loaded Structure

This paper investigates the vibration characteristics of the coupling system of a microscale fluid-loaded rectangular isotropic plate attached to a uniformly distributed mass. Previous literature has, respectively, studied the changes in the plate vibration induced by an acoustic field or by the attached mass loading. This paper investigates the issue of involving these two types of loading simultaneously. Based on Lamb's assumption of the fluid-loaded structure and the Rayleigh–Ritz energy method, this paper presents an analytical solution for the natural frequencies and mode shapes of the coupling system. Numerical results for microplates with different types of boundary conditions have also been obtained and compared with experimental and numerical results from previous literature. The theoretical model and novel analytical solution are of particular interest in the design of microplate-based biosensing devices.

Free Transverse Vibrations of Orthotropic Composite Mindlin Plates or Panels With a Non-Centrally Bonded Symmetric Lap Joint (or Symmetric Doubler Joint)

2006 ◽  
Author(s):  
U. Yuceoglu ◽  
O. Gu¨vendik ◽  
V. O¨zerciyes
Keyword(s):  
Natural Frequencies ◽  
Adhesive Layer ◽  
Mode Shapes ◽  
Shear Stresses ◽  
Lap Joint ◽  
Orthotropic Composite ◽  
Transverse Vibrations ◽  
Stress Resultant ◽  
Bonded Joint ◽  
Interpolation Polynomials

The problem of the "Free Transverse Vibrations of Orthotropic Composite Mindlin Plates or Panels with a Non-Centrally Bonded Symmetric Lap Joint (or Symmetric Doubler Joint)" is theoretically analyzed and solved with some numerical results. The "Bonded Joint" system is composed of two dissimilar, orthotropic plate "adherends" non-centrally bonded and connected by a dissimilar, orthotropic "doubler" plate through a very thin and elastic adhesive layer. The "adherends" and the single "doubler" are taken into account as the "Mindlin Plates" with the transverse shear deformations and the transverse and the rotary moments of inertia. The adhesive layer is considered as a linearly elastic continuum with the transverse normal and shear stresses. The damping effects are neglected. The dynamic equations of the plate "adherends", the "doubler" plate and the adhesive layer in combination with the stress resultant-displacement expressions, after some algebraic manipulations, are finally reduced to a set of the "Governing System of the First Order Ordinary Differential Equations" in matrix form in terms of the "state vectors" of the problem. The aforementioned set of the "Governing Equations" is integrated by means of the "Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials)". Several mode shapes with their corresponding natural frequencies are presented for the "hard" and the "soft" adhesive cases. It was found that there are significant differences in mode shapes and natural frequencies corresponding to the "hard" and the "soft" adhesive cases. Additionally, some parametric studies such as the effects of the "Bonded Joint Length Ratio" and the "Bonded Joint Position Ratio" on the natural frequencies are included in this first study.

Axisymmetric Flexural Vibrations of a Thick Free Circular Plate

1979 ◽  
Vol 46 (1) ◽  
pp. 139-144 ◽  
Author(s):  
J. R. Hutchinson
Keyword(s):  
Approximate Solution ◽  
Series Solution ◽  
Natural Frequencies ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Approximate Theory ◽  
Circular Plates ◽  
Thick Plates ◽  
Inertia Effects ◽  
Flexural Vibrations

A series solution of the general three-dimensional equations of linear elasticity is used to find accurate natural frequencies and mode shapes for the flexural vibrations of thick free circular plates. The approximate solution for thick plates, which includes shear and rotary inertia effects, is compared with the accurate series solution. It is found that the approximate solution yields frequencies of sufficient accuracy for most engineering applications within the range of applicability of the approximate theory.

On Non-Axisymmetrical Transverse Vibrations of Circular Plates of Convex Parabolic Thickness Variation

2004 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Factorization Method ◽  
Mode Shapes ◽  
Circular Plates ◽  
Partial Differential ◽  
First Order

This paper presents an approach for finding the solution of the partial differential equation of motion of the non-axisymmetrical transverse vibrations of axisymmetrical circular plates of convex parabolical thickness. This approach employed both the method of multiple scales and the factorization method for solving the governing partial differential equation. The solution has been assumed to be harmonic angular-dependent. Using the method of multiple scales, the partial differential equation has been reduced to two simpler partial differential equations which can be analytically solved and which represent two levels of approximation. Solving them, the solution resulted as first-order approximation of the exact solution. Using the factorization method, the first differential equation, homogeneous and consisting of fourth-order spatial-dependent and second-order time-dependent operators, led to a general solution in terms of hypergeometric functions. Along with given boundary conditions, the first differential equation and the second differential equation, which was nonhomogeneous, gave respectively so-called zero-order and first-order approximations of the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the first-order approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

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